A negative feedback loop in the GPCR pathway underlies efficient coding of external stimuli

Abstract Efficient navigation based on chemical cues is an essential feature shared by all animals. These cues may be encountered in complex spatiotemporal patterns and with orders of magnitude varying intensities. Nevertheless, sensory neurons accurately extract the relevant information from such perplexing signals. Here, we show how a single sensory neuron in Caenorhabditis elegans animals can cell‐autonomously encode complex stimulus patterns composed of instantaneous sharp changes and of slowly changing continuous gradients. This encoding relies on a simple negative feedback in the G‐protein‐coupled receptor (GPCR) signaling pathway in which TAX‐6/Calcineurin plays a key role in mediating the feedback inhibition. This negative feedback supports several important coding features that underlie an efficient navigation strategy, including exact adaptation and adaptation to the magnitude of the gradient's first derivative. A simple mathematical model explains the fine neural dynamics of both wild‐type and tax‐6 mutant animals, further highlighting how the calcium‐dependent activity of TAX‐6/Calcineurin dictates GPCR inhibition and response dynamics. As GPCRs are ubiquitously expressed in all sensory neurons, this mechanism may be a general solution for efficient cell‐autonomous coding of external stimuli.

stimulation pattern was observed in the AWA neuron (Larsch et al, 2015). (B) When extending the on-step period and shortening the off-step period (30 s on and 30 s off), the output activity reaches a fixed response after the first step. (C) Further increasing the frequency of the stimulus pattern (15 s on and 15 s off) leads to periodic activity skipping, where a calcium pulse is observed in response to every other on-step of the ligand. Skipping occurs because the recovery time from the inhibition ( , equations 1-4) during the short off period (between two consecutives on steps) is too short. This is an expected outcome of negative feedback loops that was also observed in the AWA neuron (Rahi et al, 2017). (D) Nearly identical output responses when doubling the concentration at each step, reflecting a typical fold-change response. (E) In response to a linear gradient, the model outcome is a series of pulses whose frequency and amplitude decay over time. (F) In response to an exponential gradient, neural activity increases with the gradient's first derivative. Calcium dynamics in (E) and (F) is also in agreement with experimental observations (Itskovits et al, 2018). All ligand patterns start at time 0 and from a baseline concentration of 1.15μM. (G) grk-2; (H) odr-3; (I) osm-6; . Activity of tax-6 mutant animals reaches a higher amplitude and lacks exact adaptation properties. Note the high variability between and within the various strains.
Black dots denote the location of the pulses, as identified by our algorithm (see Methods), and which were used in the statistical analysis of Figure 3D,E.
Appendix Figure S4. Calcium levels in eat-16 mutant animals show exact adaptation following an on-step stimulus.
(A) Neural responses of eat-16 mutants ( = 14 worms) to a five-minutes step increase of diacetyl (shown on top). While amplitude response in some neurons was low, all animals showed robust increased levels following the on step. These calcium levels gradually decreased until reaching baseline levels despite the fact that the stimulus remained on. This is in contrast to tax-6 mutants that fail to return to their baseline activity (as seen in Figure 4). (B) Median neural activity of eat-16 worms in response to the step. Shaded area marks mean absolute deviation.
Appendix Figure S5. In tax-6 mutants, the response amplitude is higher than in wt animals, while the response time remains the same.
(A) Responses to steps of diacetyl in individual worms. Worms were subjected to either one long step (top panel, = 18 wt worms) or twenty short 5-seconds long steps (middle and bottom panels, = 19, 10 wt and tax-6(p675) worms respectively), followed by a 40-seconds off step and a final 5-seconds on step. Each trace is normalized to its maximal level. (B) While the pulse amplitude of tax-6(p675) mutants is higher than that of wt worms, the time to reach the peak is similar. A zoomed-in view of representative responses of wt and tax-6(p675) worms. Top, absolute values of the response. Bottom, normalized values (by its maximal levels). (C-D).
Quantification of the response time, defined as the time to reach 80% of the maximal amplitude (C) and the peak pulse amplitude (D). While the median response time is not significantly different between tax-6(p675) mutants and wt worms, the median amplitude of tax-6(p675) mutants is significantly higher (Wilcoxon rank-sum test, = 9 • 10 −5 , = 19, 10 for wt and tax-6(p675) worms respectively). (E) Comparison between the amplitudes of wt worms and tax-6(p675) mutants in the first and last steps of the short interval steps protocol (shown in (A), middle and bottom panels). In both wt and tax-6(p675) worms, the response amplitude to the last step was lower (signed-rank test, = 1.3 • 10 −4 and 0.002 respectively), with tax-6(p675) mutants having a significantly higher amplitude in both the first and last steps (Wilcoxon rank-sum test, = 1.1 • 10 −4 and 6 • 10 −5 respectively). Black bars mark the median. (F) The difference between the amplitudes of the first and last steps is plotted as a function of the first step's amplitude. While wt worms show a larger amplitude difference when the response to the first step is stronger ( = 0.93, < 10 −8 ), tax-6(p675) worms do not ( = 0.22, = 0.53). This suggests a link between calcium influx and habituation in wt worms, but not in tax-6(p675) mutant worms. (G) Comparison between the neural responses of wt worms in the last step of the long-exposure protocol and the multiple-steps protocol. Mean calcium levels are higher during the multiple-steps protocol (as each step generates a new calcium wave response), and this neural activity is also stronger in the last step following the multiple-steps protocol (Wilcoxon rank-sum test, median of < 2 • 10 −4 ). This suggests that calcium is not the sole factor leading to habituation and additional inhibitory mechanisms may exist.

Appendix Figure S6. Eliminating the calcium-dependent inhibition from the model recapitulates key responses observed in tax-6 mutants, indicating that calciumindependent processes contribute to the inhibitory processes.
Simulations of the model outputs (denoted as calcium concentration, middle) and the inhibition dynamics (bottom) in response to various patterns of stimulus presentation (top). To simulate the responses of tax-6 mutants, we eliminated calcium-dependent inhibition (by setting k5 in the model to zero). This is the implicit model (based on Equations 1-4) which does not implement the dynamics of the AWA ion channels (Liu et al, 2018) as shown in Figure 6. The output features are qualitatively similar to the detailed model showing that tax-6 mutants are unable to achieve exact adaptation and habituate to past stimuli. They also lack pulsatile activity. (A) In contrast to the wt dynamics, in response to a sigmoidal gradient (top), simulated tax-6 worms fail to generate a pulsatile activity and their calcium levels remain at their maximal level for the duration of stimulus presentation, so that exact adaptation was not achieved. (B) wt worms exhibit lower responses (increased habituation) following the second on-step. In contrast, tax-6 mutants did not show exact adaptation, nor a reduced activity in response to the second on-step (no habituation).
Indeed, the inhibition remained low compared to wt worms. (C) Response dynamics of wt worms following a first long (continues line) or short (dashed line) on-step. A second on-step following a short on-step elicits calcium levels that match the response to the first step. This is due to the long off-step where inhibition is decreasing, thus allowing recovery (purple, note it partially overlaps with the blue and hence not visible). This contrasts with the attenuated response following a long on-step that results in a short recovery time (and hence inhibition remains high, blue). Notably, even though in both protocols, calcium concentrations returned to baseline levels, the longer the recovery time period, the higher the response activity to a second step (in agreement with experimental results, Figure 4A). This suggests that calcium-independent processes contribute to the inhibitory processes in the neuron. accumulation in the target chemoattractant region of three independent experimental repeats of both wt and tax-6 mutants. (C) Probability distribution of trajectories' directedness (Itskovits et al, 2018) for all three experiments, ranging from -1 (trajectory in opposite direction) to +1 (trajectory straight towards the target). Distribution of tax-6 animals is centered around zero, while the wt distribution is skewed towards positive values. (D) Probability distribution of movement angles relative to the target region. An angle was calculated for each frame and concatenated for all trajectories and experiments. While wt distribution is highly biased towards small angles (more directed towards the target), tax-6 distribution is more uniformly distributed, an expected result given their mostly circular undirected trajectories (Movies EV2-3). Appendix

Part 1. A detailed description and analysis of the mathematical model
Our model for chemo-sensation in the AWA neurons is composed of a system of time dependent differential equations. The circuit's topology (Figure 1) is analogous to the negative feedback loop circuit found in the chemosensory system of E. coli (Tu et al, 2008). We therefore capitalized on the mathematical framework developed in these studies and modified it to adjust to the known GPCR signaling pathway found in the AWA neurons of C. elegans worms.
Our model consists of five variables. The "Ligand" ( ) denotes the input stimulus (e.g., diacetyl). The Receptor Activation term ( ) represents the fraction of active ligand-bound receptors in the cell membrane. is facilitated by ligand binding and is regulated by Inhibition ( ) which represents the effect of inhibiting factors. Overall, is modeled as a sigmoidal function of the difference between the Ligand and Inhibition terms: Where 1 , 0 , 2 are positive constants. In this functional form, serves as the system's "memory" of past ligand concentrations. The logarithmic-scale coding of the ligand and the linear regulation of the negative feedback ("Inhibition" motif) are concepts borrowed from chemo-sensation models in E. coli (Shimizu et al, 2010;Yi et al, 2000;Tu et al, 2008).
facilitates a self-enhancing component ( ), which represents the VGCC and TRPV channels combined. Their dynamics can be described by: (2) = 3 ( − ) , (0,1) Thus, when crosses a critical activation threshold (0 < < 1), will grow/fall exponentially. is approximated to form a self-enhancing motif where a small calcium influx generated by the -mediated opening of TRPV channels initiates further opening and selfamplification of VGCCs, through which most of the calcium enters the cell (Larsch et al, 2015;Liu et al, 2018).
values are limited by code to the interval (0,1], to simplify a more realistic sigmoidal behavior. Its lower limit is a small (<<1) positive number, and its exact limit does not affect the model performance (not shown). Thus, produces a "two-state" switch where the neuron is either active or inactive. For this, the time constant 3 −1 , needs to be smaller than the other time constants in the system, though its specific values do not significantly affect the model performance.
In equation 3, facilitates calcium accumulation in the cell ( ): (3) = 4 − 1 ( − 0 ) Calcium enters through open channels ( ) and is removed by calcium pumps. The removal time constant, was approximated experimentally to be in the order of several seconds (Itskovits et al, 2018). 0 is the calcium level at rest so that ≥ 0 . The calcium in the cell facilitates the Inhibition term, according to equation 4: (4) = 5 ( − 0 ) + 6 − 1 (1 − ) Where 5 , 6 and are positive constants. Two components increase inhibition levels: the first depends on calcium (with the constant 5 ) and the second is calcium independent (denoted with the constant 6 ). The calcium-dependent term is consistent with homologous adaptation by G protein-coupled receptor kinases (GRKs) that phosphorylate only active receptors (which is why the term is multiplied by ). The second term depends on receptor activity only and is consistent with phosphorylation driven by second messenger-stimulated kinases (Lefkowitz, 1998).
These two terms fulfill two different features in the system. To achieve robust exact adaptation, the circuit's output needs to directly inhibit the receptor (Yi et al, 2000). Therefore, as calcium directly correlates with the synaptic output (Katz & Miledi, 1970;Llinás et al, 1976), its regulation of receptor inhibition underlies robust exact adaptation. However, as calcium levels are transient, adaptation should also depend on the degree of receptor activation to promote inhibition for as long as the stimulus is present. Our experimental results support this logic as we find that calcium levels correlate with adaptation magnitude, and this adaptation persists even after calcium removal (Figure 4). These results suggest two separate adaptation mechanisms: one that is calcium dependent and the other which relies on the receptor's activity.
The last term in equation 4 denotes inhibition removal which depends on inactive receptors only (thus multiplying by (1 − )), similar to methylation dynamics in E. coli (Shimizu et al, 2010). To close the negative feedback loop, this inhibition ( ) regulates receptor activation according to equation 1.

Part 2. Mathematical analysis of the model's steady state shows how exact adaptation is achieved
The mathematical formulation of exact adaptation requires that following a step increase in the attractant, the steady state value of the circuit output must return to its basal pre-stimulation levels. To examine this transition, we set → 1 and require = = = 0. From equation 2, a stable equilibrium can only be reached by = 1 or → 0. A steady state in which = 1 represents a non-pulsatile response, in which calcium stays at its maximum, and can be obtained by choices of parameters where calcium-mediated inhibition is low (as in tax-6 mutants, Appendix Figure   S6). A steady state in which → 0 may lead to exact adaptation as follows: by plugging = 0 in equation 3, we get a steady-state calcium level that does not depend on the input: = 0 .
Furthermore, plugging these results in equation 4, yields: (5) = 6 1 − If we plug in equation 1, we get a closed form for the receptor activation during steady state: This relation can be solved numerically, and by plugging this solution back to equation 5, we get the steady state inhibition level, . We can see that steady-state activation and inhibition do depend on the input level 1 , thus serving as a 'memory' of the input, even after output returns to basal levels.
Considering dynamics, when responding to a step-function input, receptor activation will rise above , elicit an output pulse that will, in turn, increase inhibition. Inhibition will eventually decrease receptor activation to terminate the pulse. In the regime where > , receptor activation will rise and cross before reaching steady state, and thus elicit another pulse.
Therefore, a strict requirement for exact adaptation in this simple model is < .

Part 3. How smooth gradients of input are encoded as a pulsatile response that adapts to the gradient's first derivative?
Our model translates smooth gradients of the input into a pulsatile response output: Increased ligand levels lead to receptor activation (eq. 1). At this stage, since there is still no output, inhibition is relatively low (eq. 4), and for simplification, we assume its effect on receptor activation to be negligible. When receptor activation reaches its threshold level, , a pulse of calcium is generated (eqs. 2 and 3). High levels of calcium rapidly increase inhibition (eq. 4), though at this time scale, ligand concentrations remain relatively constant. The rapid increase in inhibition causes a sharp decline in receptor activation levels (eq. 1), and when decreased below , the pulse terminates (eq. 2,3). Calcium is being removed and hence calcium-mediated inhibition slowly decreases (eq. 4), while ligand concentrations increase to elicit another pulse (as can be seen in Figure 2B).
To explain how the above pulsatile response adapts to the first derivative of the input gradient, we analyze the model with the following assumptions: (1) a calcium pulse is an immediate event, in which ligand levels remain constant, and the inhibition rises to a new higher level; (2) the change in inhibition during a pulse always results in a constant decrease of receptor activation; (3) between pulses, inhibition remains constant, so that only the ligand can change receptor activation.
We assume a first pulse occurs when activation reaches at a ligand level 1 and inhibition level 1 . This pulse increases inhibition to a new level, 2 , so that receptor activation falls to a new level 0 < : To elicit an additional pulse, the ligand concentration has to rise to a new level, 2 , so that receptor activation reaches once again. Inhibition is assumed to stay constant, thus when a second pulse is produced: Isolating 2 in equation 7, and placing in equation 8 yields: 10 1 is greater than 1 since > 0 and 1 > 0. Thus, to produce the consecutive pulse, ligand levels need to increase by a factor . For a linear increase in ligand levels, = , this means that the n'th pulse time, and the time between consecutive pulses will be: (10) = 1 −1 ; = Thus, the time interval between consecutive pulses grows with the number of pulses. The same consideration will also dictate less pulses in the second half of a sigmoidal gradient (decreasing first derivatives of the gradient), effectively exhibiting adaptation to the first derivative.
Part 4. Integrating the model with known dynamics of voltage gated ion channels in AWA.
To accurately account for the main voltage-gated ion channels in the AWA neuron, we integrated our model with a model developed in the Bargmann lab based on intracellular recordings of the AWA neuron (Liu et al, 2018). This model takes electrical current influx as an input and simulates the dynamics of ion channels and membrane potential in the cell. To integrate this model into our simulations, the electrical current influx was considered to be proportional to our self-enhancing motif: (2.1) = Where is the current influx, is the self-enhancing motif, and is the proportion constant which determines the connectivity between the models. Thus, the current influx is provided as the input to the model: Where is the model, is the membrane potential and is the calcium influx entering the cell.
Calcium influx is therefore plugged into equation 3, instead of : Where is a constant that changes the electrical current units to concentration changes while accounting for the volume of the AWA neuron. To determine this constant, we approximated the cell volume based on our microscopy images to be: = 9 • 4 • 4 3 ∼ 2 • 10 −13 , and used the following transformation:

Part 5. Comparing our model with the adaptive-threshold mechanism.
Levy and Bergmann (Levy & Bargmann, 2020) proposed the adaptive threshold mechanism for the timing at which the AWC neurons in C. elegans are activated in response to a chemical cue. Here we show that this model can be viewed as mathematically analogous to our model (with minor assumptions) which also provides the initiation timing of the pulses in response to stimulus gradients.
According to the adaptive threshold mechanism, the threshold is not fixed, but rather dynamically changes depending on the history of the perceived stimulus. To show how this can be equivalent to our model, we consider a smooth slowly-increasing gradient, for which we wish to calculate the concentration threshold required for the stimulus to cross in order to initiate a pulse. In our model, the condition for pulse initiation is that the fraction of active receptors crosses a threshold (eq. 2): > is a sigmoidal function of the difference between the Ligand term and the inhibition term (eq. For simplicity, we will use Taylor approximation to expand the sigmoid function for a small , such that the two terms of the right-hand equation are of the same magnitude, roughly canceling one another: (1 + (− )) −1 ≈ 1 2 + 4 , ∈ [0,1] For smooth increasing gradients, before pulse initiation, rises monotonically. Moreover, will not reach saturation (i.e., = 1), because a pulse will be initiated once crosses ( < 1).
Thus, we continue to consider the linear regime of the function. With this simplification, we get: ( ) ≈ 1 4 ( 1 log ( 0 ) − 2 ) + 1 2 Next, when considering the inhibition dynamics (eq. 4), we make the following assumptions: (1) before the initiation of the first pulse, or between consecutive pulses, calcium levels are close to baseline levels such that they do not contribute to the inhibition.
(2) The inhibition removal term, which is a function of the non-active receptors, is negligible since its time constant is an order of magnitude larger (300 sec) than the inter-pulse time (~10 sec). This